Chaotic inflation based on a simple monomial scalar potential, $V(\ensuremath{\phi})\ensuremath{\propto}{\ensuremath{\phi}}^{p}$, is an attractive large-field model of inflation capable of generating a sizable tensor-to-scalar ratio $r$. Therefore, assuming that future cosmic microwave background observations will confirm the large $r$ value reported by BICEP2, it is important to determine what kind of dynamical mechanism could possibly endow the inflaton field with such a simple effective potential. In this paper, we answer this question in the context of field theory, i.e. in the framework of dynamical chaotic inflation, where strongly interacting supersymmetric gauge dynamics around the scale of grand unification dynamically generate a fractional power-law potential via the quantum effect of dimensional transmutation. In constructing explicit models, we significantly extend our previous work, as we now consider a large variety of possible underlying gauge dynamics and relax our conditions on the field content of the model. This allows us to realize almost arbitrary rational values for the power $p$ in the inflaton potential. The present paper may hence be regarded as a first step toward a more complete theory of dynamical chaotic inflation.